Springer Solid-State Sciences

I Background.- 1. Why Study Random Matrices?.- 1.1 Statistics of the Eigenvalues of Random Matrices.- 1.1.1 Nuclear Physics.- 1.1.2 Stability of Large Ecosystems.- 1.1.3 Disordered Harmonic Solids.- 1.2 Products of Random Matrices in Chaotic and Disordered Systems.- 1.2.1 Chaotic Systems.- 1.2.2 Disordered Systems.- 1.3 Some Remarks on the Calculation of the Lyapunov Exponent of PRM.- 2. Lyapunov Exponents for PRM.- 2.1 Asymptotic Limits: the Furstenberg and Oseledec Theorems.- 2.2 Generalized Lyapunov Exponents.- 2.3 Numerical Methods for the Computation of Lyapunov Exponents.- 2.4 Analytic Results.- 2.4.1 Weak Disorder Expansion.- 2.4.2 Replica Trick.- 2.4.3 Microcanonical Method.- II Applications.- 3. Chaotic Dynamical Systems.- 3.1 Random Matrices and Deterministic Chaos.- 3.1.1 The Independent RM Approximation.- 3.1.2 Independent RM Approximation: Perturbative Approach.- 3.1.3 Beyond the Independent RM Approximation.- 3.2 CLE for High Dimensional Dynamical Systems.- 4. Disordered Systems.- 4.1 One-Dimensional Ising Model and Transfer Matrices.- 4.2 Random One-Dimensional Ising Models.- 4.2.1 Ising Chain with Random Field.- 4.2.2 Ising Chain with Random Coupling.- 4.3 Generalized Lyapunov Exponents and Free Energy Fluctuations.- 4.4 Correlation Functions and Random Matrices.- 4.5 Two-and Three-Dimensional Systems.- 5. Localization.- 5.1 Localization in One-Dimensional Systems.- 5.1.1 Exponential Growth and Localization: The Borland Conjecture.- 5.1.2 Density of States in One-Dimensional Systems.- 5.1.3 Conductivity and Lyapunov Exponents: The Landauer Formula.- 5.2 PRMs and One-Dimensional Localization: Some Applications.- 5.2.1 Weak Disorder Expansion.- 5.2.2 Replica Trick and Microcanonical Approximation.- 5.2.3 Generalized Localization Lengths.- 5.2.4 Random Potentials with Extended States.- 5.3 PRMs and Localization in Two and Three Dimensions.- 5.4 Maximum Entropy Approach to the Conductance Fluctuations.- III Miscellany.- 6. Other Applications.- 6.1 Propagation of Light in Random Media.- 6.1.1 Media with Random Optical Index.- 6.1.2 Randomly Deformed Optical Waveguide.- 6.2 Random Magnetic Dynamos.- 6.3 Image Compression.- 6.3.1 Iterated Function System.- 6.3.2 Determination of the IFS Code for Image Compression.- 7. Appendices.- 7.1 Statistics of the Eigenvalues of Real Random Asymmetric Matrices.- 7.2 Program for the Computation of the Lyapunov Spectrum.- 7.3 Poincare Section.- 7.4 Markov Chain and Shannon Entropy.- 7.5 Kolmogorov-Sinai and Topological Entropies.- 7.6 Generalized Fractal Dimensions and Multifractals.- 7.7 Localization in Correlated Random Potentials.- References.